Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

Abstract

Symmetry-protected topological phases (SPT) are short-range entangled gapped\nstates protected by global symmetry. Nontrivial SPT phases cannot be\nadiabatically connected to the trivial disordered state(or atomic insulator) as\nlong as certain global symmetry $G$ is unbroken. At low energies, most of\ntwo-dimensional SPTs with Abelian symmetry can be described by topological\nquantum field theory (TQFT) of multi-component Chern-Simons type. However, in\ncontrast to the fractional quantum Hall effect where TQFT can give rise to\ninteresting bulk anyons, TQFT for SPTs only supports trivial bulk excitations.\nThe essential question in TQFT descriptions for SPTs is to understand how the\nglobal symmetry is implemented in the partition function. In this paper, we\nsystematically study TQFT of three-dimensional SPTs with unitary Abelian\nsymmetry (e.g., $\\mathbb{Z}_{N_1}\\times\\mathbb{Z}_{N_2}\\times\\cdots$). In\naddition to the usual multi-component $BF$ topological term at level-$1$, we\nfind that there are new topological terms with quantized coefficients (e.g.,\n$a^1\\wedge a^2\\wedge d a^2$ and $a^1\\wedge a^2\\wedge a^3\\wedge a^4$) in TQFT\nactions, where $a^{1},a^2,\\cdots$ are 1-form U(1) gauge fields. These\nadditional topological terms cannot be adiabatically turned off as long as $G$\nis unbroken. By investigating symmetry transformations for the TQFT partition\nfunction, we end up with the classification of SPTs that is consistent with the\nwell-known group cohomology approach. We also discuss how to gauge the global\nsymmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.\n